Question

Write the complex number (1−i)^(1+i )in standard form a+ib where i=√−1is the imaginary unit.

Answer 1

write exponential form

1−i=√2(√22+i−√22)=√2e−iπ4

Substitute 1−i by the above in the given expression (1−i)1+i

(1−i)1+i=(√2e−iπ4)1+i

Use exponents rule to take √2 from inside the brackets

(1−i)1+i=(√2)1+i(e−iπ4)(1+i)

Use exponents rule to rewrite as

(1−i)1+i=√2eπ4√2ie−iπ4

√2 may be written as eln√2 ; hence the given expression may be written as

(1−i)1+i=√2eπ4(eln√2)ie−iπ4

Simplify

(1−i)1+i=√2eπ4e(ln√2−π4)i

Write in standard form

(1−i)1+i=√2eπ4cos(ln√2−π4)+i√2eπ4sin(ln√2−π4)

 (1-i)^(1+i)